A238795 -id:A238795 - cp's OEIS frontend

A239036 A set of eleven distinct positive odd numbers the sum of whose reciprocals is 1 and whose 11th term is as large as possible.

3, 5, 7, 9, 11, 13, 23, 721, 979011, 175878510309, 20622166925499467673345

Offset: 1

Arkadiusz Wesolowski, Mar 09 2014

If k is the largest number in the set of eleven distinct positive odd numbers the sum of whose reciprocals is 1, then k <= a(11).
Is there any set of eleven distinct positive odd numbers the sum of whose reciprocals is 1 and having the Egyptian number greater than 20622166925675347163457?
This is similar to the problem discussed by Curtiss (see link), but the numbers are restricted to be odd. - T. D. Noe, Mar 18 2014

			1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/13 + 1/23 + 1/721 + 1/979011 + 1/175878510309 + 1/20622166925499467673345 = 1.

D. R. Curtiss, On Kellogg's Diophantine problem, Amer. Math. Monthly 29 (1922), pp. 380-387.
Index entries for sequences related to Egyptian fractions

Cf. A238795, A201646.

f=0; n=3; s=11; if(s<11, break); for(t=1, s-3, print1(n, ", "); f=f+1/n; until(1>f+1/n, n=n+2)); until(numerator(1-f-1/n)==2, n=n+2); print1(n, ", "); f=f+1/n; g=2*floor((numerator(f)+1)/4)+1; until(numerator(1-f-1/g)==1, g=g+2); print1(g, ", "); f=f+1/g; print1(denominator(1-f));

A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 5, 0, 0, 1, 2, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 5, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 7, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 2, 3

Offset: 1

Lars Blomberg, Feb 14 2022

nonn

By symmetry, if (i, j) is a solution then so is (j, i). When j=i we get n = 3i, corresponding to the solution 1/2 + 1/2 = 1. Therefore, when 3|n, a(n) > 0 and odd, otherwise a(n) >= 0 and even.
For n < 10^6, the largest term is a(720720) = 285, and 483188 terms are 0.
Other record terms: a(1081080) = 369, a(2162160) = 457, a(3243240) = 481, a(4324320) = 533, a(5405400) = 559, a(6126120) = 597. Record terms with index >= 360360 appear to occur at indices that are multiples of 180180. - Chai Wah Wu, Feb 15 2022

			For n = 3: (i, j) = (1, 1), so a(3) = 1. (1/2 + 1/2 = 1)
For n = 18: (i, j) = (3, 8), (6, 6), (8, 3), so a(18) = 3. (3/15 + 8/10 = 1/5 + 4/5 = 1)
For n = 20: (i, j) = (5, 8), (8, 5), so a(20) = 2.
For n = 36: (i, j) = (6, 16), (8, 15), (12, 12), (15, 8), (16, 6), so a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A018892, A016017, A051908, A073546, A227610, A238795, A241883, A297895, A303400.

Cf. A351694, A351695 for records.

a(n)={my(x=n^2, y=2*n); sum(i=1,(n-1)/2, x-=2*n; y-=3; if(x%y==0,1,0))}

from sympy.abc import i, j
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A351532(n):
    return sum(1 for d in diop_quadratic(n**2+3*i*j-2*n*(i+j)) if 0 < d[0] < n and 0 < d[1] < n) # Chai Wah Wu, Feb 15 2022

The original equation can be solved for j giving j = (n(n - 2i)) / (2n - 3i). Varying i from 1 to n-1, a(n) is given by the number of integer values of j, 0 < j < n.

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

A240132 Number of solutions of 1/x_1 + 1/x_2 + ... + 1/x_{2n-1} = 1 in distinct odd numbers > 1.

Original entry on oeis.org

0, 0, 0, 0, 5, 379118

Offset: 1

Jonathan Sondow, Apr 21 2014

The next term is a(7) > 467871307.

			If 1 < x_1 < x_2 < x_3 are odd, then 1/x_1 + 1/x_2 + 1/x_3 < 3/3 = 1, so a(2) = 0.

R. Arce-Nazario, F. Castro, and R. Figueroa, On the number of solutions of Sum_{i = 1..11} 1/x_i = 1 in distinct odd natural numbers, J. Number Theory, 133 (2013), 2036-2046.
C. Elsholtz, Review of ``On the number of solutions of sum_{i = 1..11} 1/x_i = 1 in distinct odd natural numbers'' by R. Arce-Nazario, F. Castro, and R. Figueroa, zbMATH 1270.11033.

The five sets of 9 odd numbers are: A201644, A201646, A201647, A201648, A201649.

See A238795 and A239036 for specific sets of 11 odd numbers.

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A239036 A set of eleven distinct positive odd numbers the sum of whose reciprocals is 1 and whose 11th term is as large as possible.

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A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

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A240132 Number of solutions of 1/x_1 + 1/x_2 + ... + 1/x_{2n-1} = 1 in distinct odd numbers > 1.

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